Identities involving Narayana numbers

Narayana’s cows problem is a problem similar to the Fibonacci’s rabbit problem. We define the numbers which are the solutions of this problem as Narayana’s cows numbers. Narayana’s cows sequence satisfies the third order recurrence relation $N_r=N_{r-1}+N_{r-3}$ ($r\ge 3$) with initial condition $N_0=0$, $N_1=N_2=1$. In this paper, the $ar+b$ subscripted Narayana numbers will be expressed by three $a$ step apart Narayana numbers for any $1\le b\le a$ ($a\in \mathbb{Z}$). Furthermore, we study the sum $S_{N,r}^{(4,b)}={\sum }_{k=0}^r{N}_{4k+b}$ of $4$ step apart Narayana numbers for any $1\le b\le 4$.